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Issue ESAIM: PS
Volume 5, 2001
Page(s) 33 - 49
DOI 10.1051/ps:2001101

DOI: 10.1051/ps:2001101


ESAIM: P&S, July 2001, Vol. 5, pp. 33-49

Model selection for (auto-)regression with dependent data

Yannick Baraud1, F. Comte2 and G. Viennet3

1  École Normale Supérieure, DMA, 45 rue d'Ulm, 75230 Paris Cedex 05, France; (Yannick.Baraud@ens.fr)
2  Laboratoire de Probabilités et Modèles Aléatoires, Boîte 188, Université Paris 6, 4 place Jussieu, 75252 Paris Cedex 05, France.
3  Laboratoire de Probabilités et Modèles Aléatoires, Boîte 7012, Université Paris 7, 2 place Jussieu, 75251 Paris Cedex 05, France.

(Received April 15, 1999. Revised July 20, 1999 and May 14, 2001.)

Abstract
In this paper, we study the problem of non parametric estimation of an unknown regression function from dependent data with sub-Gaussian errors. As a particular case, we handle the autoregressive framework. For this purpose, we consider a collection of finite dimensional linear spaces (e.g. linear spaces spanned by wavelets or piecewise polynomials on a possibly irregular grid) and we estimate the regression function by a least-squares estimator built on a data driven selected linear space among the collection. This data driven choice is performed via the minimization of a penalized criterion akin to the Mallows' Cp. We state non asymptotic risk bounds for our estimator in some $\L _2$-norm and we show that it is adaptive in the minimax sense over a large class of Besov balls of the form ${\mathcal B}_{\alpha,p,\infty}(R)$ with $p\geq 1$.


AMS Subject: 62G08, 62J02.

Key words: Nonparametric regression, least-squares estimator, adaptive estimation, autoregression, mixing processes.


© EDP Sciences, SMAI 2001


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